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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.21029 |
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Table of Contents:
- The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\left(\frac{\partial}{\partial t} - Δ\right) u \leq A u^p + B u $$ with small initial data in borderline Morrey norms over a Riemannian manifold with bounded geometry. We obtain $L^\infty$ estimates assuming $$\|u(\cdot,0)\|_{M^{q, \frac{2q}{p-1}}} + \sup_{0 \leq t < T} \|u(\cdot, t) \|_{L^s} < δ,$$ where $1 < q \leq q_c := \frac{n(p-1)}{2}$ and $1 \leq s \leq q_c$. Assuming also a bound on $\|u(\cdot, 0)\|_{M^{q', λ'}}$, where either $q' > q$ or $λ' < \frac{2q}{p-1}$, we get an improved estimate near the initial time. These results have applications to geometric flows in higher dimensions.